Various structures in graph theory and number theory capture pseudo-randomness by small excluded substructures. Conversely, truly random matrices introduce statistical structure (e.g., Wigner's Semi-Circle Law). The proposal addresses these general phenomena through three specific classes of problems: graph problems with certain excluded subgraphs (small cliques or trees), number theory problems about sum-free sequences of integers, and various structural theorems for random matrices.
The proposed topics are applicable in mathematics and the sciences. In particular, the graph problems may find applications in Fourier analysis (and hence in Communications Technology), in Discrete Geometry (and hence in Robotics), in Number Theory (and hence in Coding Theory), and in designing and analysing efficient computer algorithms (Complexity Theory). Random matrices have their most notable applications in Mathematical Physics, but they are also very important in a broad range of sciences and several branches of mathematics from Multivariate Statistics to Operations Research to Bioinformatics to Learning Theory -- just to mention some important ones.
